Post 13: Perverse Sheaves Part 6.5

17 Jan 2024

This post fills in some technical details left out from Part 6. If you are just reading these posts for the intuition, you don’t need to read this.

Injective objects

I mentioned injective sheaves last time, in the context of being a right adapted class for several left exact functors of sheaves. Let me expand upon that here.

One can define injective objects in any category. Here is the definition:

An object $Q$ is injective if for every monomorphism $f:X\to Y$ and every morphism $g:X\to Q$, there is a morphism $h:Y\to Q$ with $h\circ f = g$.

In case you forgot, a monomorphism is the categorical analogue of injective functions. They are defined by left-cancellability: if $f\circ g = f\circ h$, then $g=h$.

A category has “enough injectives” if for every object $X$, there is a monomorphism $X\to Q$ with $Q$ injective.

The key fact is this:

If an abelian category $\mc A$ has enough injectives, then the injective objects are a right adapted class for any left exact functor $\mc A\to \mc B$. In particular, every left exact functor has a right derived functor.

There is also a notion of projective objects which is defined in a “dual” way and plays the same role for right exact functors.

In our context, an important fact is that $Sh(X)$ has enough injectives, so any left exact functor has a right derived functor. However, $Sh(X)$ does not always have enough projectives, so in order to find left derived functors, one has to find a more specific adapted class.

Sheaf functors induced by a continuous map

I did not define any of the functors $^\circ f_*$, $^\circ f^*$, $^\circ f_!$, despite claiming that they were simpler than $f^!$. Before we do that, let me cover my bases and give some necessary definitions.

Topological prerequisite: proper maps

Before we can define the functors, I need to define some notions you might not have seen before.

A continuous map $f:X\to Y$ is closed if for all closed subsets $C\subset X$, the set $f(C)$ is closed.
A continuous map $f:X\to Y$ is proper if for all spaces $Z$, the map $f\times 1_Z:X\times Z\to Y\times Z$ defined by $(x,z)\mapsto (f(x),z)$ is closed.

When $X$ and $Y$ are sufficiently nice, in particular locally compact, then the following two conditions on $f$ are individually equivalent to $f$ being proper:

For every compact subset $K\subset Y$, the set $f^{-1}(K)$ is compact.
$f$ is closed, and for every $y\in Y$, the set $f^{-1}(y)$ is compact.

I won’t discuss local compactness, but I will emphasize that most, if not all, of the spaces one deals with in practice in geometry are locally compact, so you can think of a map being proper in terms of either of the above conditions.

We also need the following notion. Let $F$ be a presheaf on $X$, and let $s\in F(U)$ be a section. Recall that for all points $p\in U$, there is a natural map $F(U)\to F_p$. Let the image of $s$ under this map be $s_p$. Then the support of $s$ is the set of points $p$ where $s_p\neq 0$. It is denoted by supp $s$.

Categorical prerequisite: colimits

We also need the notion of colimit. Limits and colimits are defined by universal properties, although just like all objects defined by universal properties, it need not exist in a particular category.

Let $F:\mc J\to \mc C$ be a functor. A colimit of $F$ is an initial object in the following category:

Objects are pairs $(X,f)$ consisting of an object $X\in \mc C$ and a collection $f$ of morphisms $f_J:F(J)\to X$ for each $J\in \mc J$ with the additional property that if $j:J\to J’$ is a morphism in $J$, then $f_{J’}\circ F(j)=f_J$.
A morphism $h:(X,f)\to (Y,g)$ is a morphism $h:X\to Y$ such that $g_J=h\circ f_J$ for each $J\in \mc J$.

If $(L,f)$ is a colimit of $F$, we sometimes also just call $L$ a colimit and call the morphisms $f_J$ the “canonical” morphisms to $L$.

Onto the definitions

We are now ready to define the three sheaf functors induced by $f:X\to Y$.

For $F\in Sh(X)$, we define $^\circ f_*(F)\in Sh(Y)$ by \[^\circ f_*(F)(U)=F(f^{-1}(U)).\]
For $F\in Sh(X)$, we define $^\circ f_!(F)\in Sh(Y)$ by \[^\circ f_!(F)(U)=\{s\in F(f^{-1}(U))\mid f|_{\text{supp }s}:\text{supp }s\to U\text{ is proper } \}.\]
For $F\in Sh(Y)$, we first define a presheaf $f^*_{pre}(F)$ on $X$ as follows. For each open $U\subset X$, let $F_U$ be the functor $F$ restricted to the full subcategory of $Op(Y)$ consisting of opens $V\subset Y$ that contain $f(U)$. Then, let $f^*_{pre}(F)(U)=\text{colim }F_U$. Finally, we define $^\circ f^*(F)$ to be the sheafification of $f^*_{pre}(F)$.

I should mention that there is actually a lot that I’m leaving out here; for instance, functors send objects to objects and morphisms to morphisms, but I’m only saying how they are defined on objects. However, this is rather typical in math exposition, especially if there is a “natural” way to define how the functors act on morphisms. I actually hide this exact issue twice: once for the sheaf functors $^\circ f$, and again for the outputs $^\circ f(F)$.

Footnotes: